February 13, 2012 1 Comment
Moreover, the isomorphism is natural in both and .
Wow that looks complicated. Let’s parse some of the notation that I haven’t even explained yet. So certainly we know what the righthand-side means. It’s applied to . That’s just a set. As for the lefthand-side, and I haven’t explained.
So is a functor we mentioned briefly, but I used a different notation. It’s the representable functor . I write it as here to avoid over using parentheses and therefore complicating this business well beyond it’s current level of complication. As a reminder, is a functor from to (the same as ). It takes objects in to the set of homomorphisms . Remember we’re in a locally small category, so really is a set. It takes morphisms to a map by sending
We checked all the necessary details in an earlier post to make sure this really was a functor.
So the last thing is that . It’s the collection of all natural transformations between the functors and . So Yoneda’s lemma claims that there is a one to one correspondence between the the natural transformations from to and the set .
In particular, it claims that is a set. This is because is a locally small category, and so each natural transformation (defined as a collection of morphisms, each of which was a set) is a set. But there are only so many collections of morphisms, not even all of which are natural transformations. The collection is small enough to be a set. If you don’t care about this set theory business. Then disregard the paragraph you probably just read angrily.
It’s worth mentioning now, that Yoneda’s lemma is a generalization of some nice theorems. We can (and will) use it to derive Cayley’s theorem (every group embeds into a symmetric group). We can (and will) use it to derive the important fact that in the category of -modules. I bet in that one you can already start to see the resemblance.
We’ll prove Yoneda’s lemma over the next few posts.